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78 3-L DC Link ML Converter Properties
4.8 NP current as a function of input current and switching function
harmonics
The NP current is a function of output currents and the switching functions of the inverter as
shown in (45). The same relation ship is true if the instantaneous switching function s x(t) is used.
i
NPx
( t)
= i ( t)*(1
− abs(
s ( t)))
(69)
x
x
To understand what impact harmonics in load current and in the switching function have, a
relationship in the frequency domain is required. However, this is very complex, as there is no
simple transformation of the absolute value function into the frequency domain. Phase and
amplitude of all individual harmonics have an impact on the presence of zero crossings of s x(t) and
therefore not just quantitatively but also qualitatively influence the absolute value function. A
simple analytically closed solution has not been published so far and does not seem to be possible.
A few papers assume harmonics in the load current but sinusoidal modulation, which is
mathematically less complex but still meaningful in practice. [104] determines the relevant current
harmonics (non-triple even harmonics) generating a DC NP current. [92] gives some curves for the
impact of specific harmonics (2 nd and 4 th ) in the load current on the NP DC current. Marchesoni
proposes a NP control schemes based on load harmonics [91] and needs a relationship between
load harmonics and NP currents for that purpose. He too uses sinusoidal modulation and
approximations due to the lack of precise analytical solutions. Pou [105] analyzes the impact of
linear but unbalances systems (load currents composed only of positive and negative sequence
fundamental components) as well as nonlinear load (specifically with 2 nd and 4 th order harmonics).
As a result, he determines the limits of 2 nd and 4 th harmonic content for stable operation using
SVM. Although he states the NP current in function of output currents and switching states, he
does not need to establish a relationship between the spectrum of the switching function and the
spectrum of the load current for his purpose.
Scheuer [106] proposes to use the square function in (70) to approximate (69). For a function s
taking exclusively the values -1, 0 and 1 as would be the case for the discrete states in a 3-L
converter, (69) and (70) are equal. This approximation can therefore be used if the exact switching
pattern is used (no averaging) in the case of 3-L converters.
i
NPx
2 x
t
( t)
= i ( t) *(1 − s ( ))
(70)
x
In contrast to the absolute value function the spectrum of s x
2(t) can be calculated. The resulting
spectrum can then be combined with the input current spectrum to obtain the NP current
spectrum. However, the two functions do not correspond if s x(t) assumes any other value than -1, 0
or 1. In the case of the 5-L converter s x(t) can assume also -0.5 and +0.5, resulting in a poor
approximation of the true NP current. An approximation of (45) using the square of the average
switching function is not possible at all as s x
(t)
is a continuous function from -1 to 1. Therefore
square approximation according to (70) is not used in this document.
Another type of simplification is proposed below, giving exact result in the considered
operating range. If there is no DC offset and only a single zero crossing per half period, the